# With a little help from Pythagoras

One way of converting 1 to 0 and 0 to 1 is to put Pythagoras theorem in a sphere and let it spin. Goes like this:

a^2 + b^=c^
Let a be the axis of the sphere, and b the radius.
Let axis a be a vertical 1 and b the 0 horizon.
Let a decrease from 1 to 0 Let b increase from 0 to 1

1^2+0^2           =1               sqrt(1) = 1
-.75^2+.25^2   =-.5             sqrt(-.5) = 0.7071067811865475i
-.5^2+.5^2       =0                sqrt(0) = 0
-.25^2+.75^2   =.5              sqrt(.5) = 0.7071067811865475
0^1+1^2          =1                sqrt(1) = 1

So what’s so special with this sequence? Well, what’s special to me might be irrelevant to you so it’s your call really. To me it says:
Imaginary/Invisible spherical equilibrium of 1
is less invisible as the 1 imaginary axis is compressed to -.75 by horizontal tension
and when axial compression -.5 equals horizontal ex-tension .5, there is energized equilibrium which increases in empirical visibility
to the extent where 0 imaginary aspect is to be found at horizon 1.

…and by this, we have generated a real plane out of an imaginary sphere, so 1i has translated itself to a real 1. As I have suggested in other posts, this is not some end state of the function. Axial compression, all the way to 0 “length” does not imply there is something lost of the unit. Spatial length is only horizontal, not vertical. We can assign “gravitational potential” to the axis and know that it is 0 at the moment of sphere, and 1 at the moment of disc.
That being potentially so, the above sequence will invert at c^2 = real 1, and the functional unit will then become increasingly smaller and de-charged. That is how reality is quantized by self-generated “gravity”.

I can also see that there are two distinct forms of equilibrium. The conventional equilibrium would be at -.5^2+.5^2=0. That is, when kinetic energy .5 equals potential energy -.5. This is the equilibrium in the classical world. But the classical world is not of a single unit as in this case. The classical world is of parity, and in a many body system, one thing absolute is, by definition, everything relative. So the other equilibrium, and in my opinion the far more interesting one, is the spherical one where there is no energy at all. This is a grey state of neither extension, nor compression. It is essentially dead to the empirical world of observation. This is when 1 is a purely imaginary 1i. In relation to the mod4 of imaginary numbers we can see that whenever i is raised to the power of even numbers e.g. -4, -2, 0, 2, 4 etc, it equals 1 or -1, but raised to odd numbers e.g. -3, -1, 1, 3 etc, it equals i or –i.
So how come the real power of the imaginary part rise only to the power of equals? Well, if my model is correct, it has to be so on the most fundamental level of single quantum mechanics/functions/operations. That is because any instance of kinetic charge has at its (invisible) center an equal amount of compression. So the power of 2 comes hand in hand in any realized, empirically detectable object. That is not to say there is no power of oddities, because obviously there is . Point is that such powers are of relatives and not absolutes, so instead of 1 or -1 we get i or -i. It is real values for sure, but they do not belong to a specific unit that can be empirically defined. It is more like “field” values.

Anyways, that’s what I find tasty food for thought because it plays very well with my imagination. But others work with other ideas, and perhaps you will find other valuables than I do.

Since the number 0.7071067811865475 turned up, both as i when function is ¼ real and as real when function is ¼ imaginary, I googled it and found something slightly interesting, even for someone who knows next to nothing about math.

For an example of an operator that throws its basis Kets into superpositions, here’s a function emulating a Hadamard operator:

julia> @def_op ” h | n > = 1/√2 * ( | 0 > + (-1)^n *| 1 > )”

h (generic function with 1 methods)

julia> d” h * | 0 > ”

Ket{KroneckerDelta,1,Float64} with 2 state(s):

0.7071067811865475 | 0

0.7071067811865475 | 1

julia> d” h * | 1 > ”

Ket{KroneckerDelta,1,Float64} with 2 state(s):

0.7071067811865475 | 0 ⟩

-0.7071067811865475 | 1 ⟩

Make of it what you will. There is something of importance going on there.
That’s what I think.